A three-digit number with no repeated digits is made from the digits 1

With given conditions there are patterns that must be identified to proceed in an efficient manner. Conditions are:
1. Only 1 to 7 digits are used
2. There is no repetition of digits
3. The numbers required are between 300 and 650

For ease we divide 300 to 650 in parts i.e. 300-400, 400-500, 500-600 and 600-650. Only the last one(600-650) is different from other ranges.

300-400:
Hundred place can be taken by only one digit i.e. '3'. Hence only one digit.
Tens place can be taken by digits other than '3' i.e. 1, 2, 4, 5, 6 and 7, totalling six digits.
Ones place can be taken by digits other than '3' and one that appear at tens place, totalling 5 digits.
Therefore, total numbers = 1*6*5 = 30

400-500:
Hundred place can be taken by only one digit i.e. '4'. Hence only one digit.
Tens place can be taken by digits other than '4' i.e. 1, 2, 3, 5, 6 and 7, totalling six digits.
Ones place can be taken by digits other than '4' and one that appear at tens place, totalling 5 digits.
Therefore, total numbers = 1*6*5 = 30

Similarly, for 500-600

600-650: This will be different as its not upto 700 to give 30 numbers in this range as we got earlier.
Hundred place can be taken by only one digit i.e. '6'. Hence only one digit.
Tens place can be taken by digits other than '6' i.e. 1, 2, 3 and 4, totalling four digits.
Ones place can be taken by digits other than '6' and one that appear at tens place, totalling 5 digits.
Therefore, total numbers = 1*4*5 = 20

Total required numbers are 30 + 30 + 30 + 20 = 110

Total numbers using given conditions with digits 1 to 7 are = 7*6*5 = 210 (123 to 765 - each 100 number range having 30 required numbers)
Probability = \(\frac{110}{210} = \frac{11}{21}\)

Answer C.

Probability number is between 300-599 + probability number is between 600-650
D1 D2 D3 + D1 D2 D3
3/7 * 6/6 * 5/5 + 1/7 *4/6 *5/5

=11/21

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